Reducible boundary conditions in coupled channels

TL;DR

This paper investigates Hamiltonians with point interactions in vector-valued function spaces, identifying conditions under which they can be decomposed into simpler one-dimensional problems, with applications to quantum graph models.

Contribution

It introduces a class of reducible Hamiltonians with point interactions, linking their reducibility to invariance under channel permutations, and provides concrete examples including delta and Kirchhoff couplings.

Findings

Reduction is linked to invariance under channel permutations.

Certain point interactions can be decomposed into simpler problems.

Examples include delta, delta-prime, and Kirchhoff couplings.

Abstract

We study Hamiltonians with point interactions in spaces of vector-valued functions. Using some information from the theory of quantum graphs we describe a class of the operators which can be reduced to the direct sum of several one-dimensional problems. It shown that such reduction is closely connected with the invariance under channel permutations. Examples are provided by some "model" interactions, in particular, the so-called delta, delta-prime, and the Kirchhoff couplings.